Crystal field splitting in correlated systems with the negative charge

arXiv:1106.4648v1 [cond-mat.mtrl-sci] 23 Jun 2011

Crystal field splitting in correlated systems with the negative charge transfer gap A V Ushakov II. Physikalisches Institut, Universit¨ at zu K¨ oln, Z¨ ulpicher Straße 77, D-50937 K¨ oln, Germany E-mail: [email protected]

S V Streltsov Institute of Metal Physics, S.Kovalevskoy St. 18, 620990 Ekaterinburg, Russia Ural Federal University, Mira St. 19, 620990 Ekaterinburg, Russia E-mail: [email protected]

D I Khomskii II. Physikalisches Institut, Universit¨ at zu K¨ oln, Z¨ ulpicher Straße 77, D-50937 K¨ oln, Germany Abstract. Special features of the crystal field splitting of d−levels in the transition metal compounds with the small or negative charge-transfer gap ∆CT are considered. We show that in this case the Coulomb term and the covalent contribution to the t2g − eg splitting have different signs. In order to check the theoretical predictions we carried out the ab-initio band structure calculations for Cs2 Au2 Cl6 , in which the charge-transfer gap is negative, so that the d−electrons predominantly occupy low-lying bonding states. For these states the eg -levels lie below t2g ones, which demonstrates that at least in this case the influence of the p − d covalency on the total value of the crystal field splitting is stronger than the Coulomb interaction (which would lead to the opposite level order). We also show that the states in conduction band are made predominantly of p−states of ligands (Cl), with small admixture of d−states of Au.

Crystal field splitting in correlated systems with the negative charge transfer gap

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1. Introduction One of the most important factors determining properties of the transition metal (TM) oxides is the splitting of d-levels when the TM ions are put in a crystal. This crystalfield (CF) splitting largely determines magnetic and orbital state of respective ions, and thus their properties. In general there are two mechanisms, which have to be taken into account in the calculation of the CF splitting. One is the Coulomb interaction of d-electrons with charges of surrounding ions - mainly nearest-neighbor anions like O2− in oxides or F− and Cl− in fluorides and chlorides. Since the Coulomb repulsion with anions depends on the shape of the electron density, the resulting corrections to the energy spectrum will be different for different d−orbitals. The second mechanism which also leads to the CF splitting is the covalency, the hybridization between d−states of the TM ions and ligand p−states. The p − d hybridization (hopping integrals tpd in Eq. (1)) strongly depends on the symmetry of the local environment and will be very different for d−orbitals pointing to the ligands (eg orbitals in the case of a octahedral surrounding) or looking in between of them (t2g orbitals in the octahedra), which would lead to their splitting. Here and below we will discuss the octahedral symmetry. We show below that the role of these two mechanisms may strongly differ in conventional TM compounds and in the materials with the negative charge-transfer (CT) gaps [1, 2], with a (e.g. oxygen) holes in the ground state. 2. The basic model The standard model describing transition metal (TM) ions in a crystal is H = ǫd0 +

X j

+

X

d†iασ diασ + ǫp

iα

p†jβσ pjβσ

jβ

† tpd ijαβ (dασ pjβσ

X

X

+ h.c.)

dd Uαβ niασ niβσ′ .

(1)

(ασ)6=(βσ′ )

Here di† iασ , diασ are the creation and annihilation operators of d−electrons at a site i in the orbital state α with spin σ, p†jβσ , pjβσ describe different p−electrons at the j−th ligand surrounding given TM site. The system is also characterized by the onsite Coulomb (Hubbard) interaction U dd = U, and by the charge-transfer energy ∆CT , defined as the energy of the transition dn p6 → dn+1 p5 ,

(2)

which in the electron representation used in the Hamiltonian (1) is ∆CT = ǫd0 + nU − ǫp = ǫd − ǫp .

(3)

Crystal field splitting in correlated systems with the negative charge transfer gap

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Here we introduced the renormalized energy of d−levels ǫd = ǫd0 + nU

(4)

- the effective d−energy levels to which an electron is transferred from p−states of a ligand in the “reaction” (2). [3] Further on in this paper by d−levels shown e.g. in Figs. 1 and 3 we have in mind these renormalized d−levels (4). In principle one should include in this Hamiltonian some other terms, such as the intra-atomic Hund’s rule exchange JH , Coulomb repulsion for ligand p−electrons Upp , etc. For our purposes, however, the simplified model (1) is sufficient. 3. Positive charge-transfer energy, ∆CT > 0 In a most typical situation for the TM compounds, when the ligand p−levels lie deep below d−levels of the TM ions, both the Coulomb and the hybridization contributions to the CF splitting lead to the same sequence of d−levels. This case is characterized by a large positive CT gap ∆CT , defined in Eq. (3). As explained above and as is clearly seen in Fig. 1(a), the Coulomb repulsion of d-electrons with negatively-charged ligands is stronger for eg −electrons with the lobes of the electron density directed towards the ligands than for t2g −orbitals pointing in between the anions. This leads to the CF splitting shown in Fig. 1(c) such that eg −orbitals lie above t2g . This sequence of d−levels is referred as “normal”. The same “normal” sequence of d−levels is also caused by the p − d hybridization, i.e. by covalent contribution. Since the hopping matrix elements eg − p (tpdσ ) are larger than hoppings between t2g −orbitals and p−orbitals of ligands (tpdπ ), the eg −levels are pushed up stronger than t2g , see Figs. 1(b), (c). [4] Relative importance of these two contributions to the total value of the CF splitting depends on the compound, but in the usual situation of ∆CT > 0 they at least work in the same direction. In the early stage of the development of the CF theory it was argued that an account of the real shape of ligand orbitals may change the sign of the Coulomb contribution, obtained in the point-charge model. [5] However, it was shown later that in order to consider ionic term in a correct way the d−electron wave functions must be orthogonalized to the ligands. [6] As a result, the total ionic contribution to the CF splitting has the same sign as the covalent term in the case of ∆CT > 0. [7] Note that for the bonding levels (the lower levels in Fig. 1(c)) the covalency contribution gives the sequence of levels opposite to the “normal”: the levels with the eg −symmetry have stronger bonding-antibonding splitting and lie below those with the t2g −symmetry. But for large positive CT gap these states are predominantly made of the combination of p−states of ligands with a proper symmetry, with only a small admixture of d−states, and one rarely discuss these states, which lie deep below the Fermi energy. All interesting phenomena take place in the partially-filled antibonding states, which have predominantly d−character and which are responsible e.g. for magnetism, orbital

Crystal field splitting in correlated systems with the negative charge transfer gap

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Figure 1. (color online) Charge-transfer gap ∆CT > 0. There are two contributions to the resulting CF splitting caused by (a) the Coulomb interaction and (b) the p − d hybridization (or covalency). The eg −orbitals (the x2 − y 2 is shown as an example) pointing to the ligands have larger hybridization and stronger repulsion from them and as a result lie above the t2g −levels (the yx−orbital is sketched) . The ligands forming octahedra are shown as open circles, d−orbitals are painted in blue and ligand p−orbitals in green. For the Coulomb contribution the charges (not the signs of the wave-functions) are indicated. The levels with predominantly d−character are marked by the dashed “box”.

ordering etc. But this changes drastically in case of the negative CT gap ∆CT < 0, which we now consider. 4. Negative charge-transfer energy, ∆CT < 0 The situation is quite different for the systems with small and especially negative CT gaps. In this case the Coulomb and covalent contributions to the splitting of d−levels have different signs. The negative CT gap is met in the compounds based on the late TM ions with unusually high valence, or high oxidation state [2], like Fe4+ or Cu3+ , see Fig. 2, compiled by Mizokawa on the basis of the data of [8, 9]. The negative CT energy means that it is favorable to transfer an electron from the ligand p-shell to a d-level, i.e. to perform the “reaction” (2) already in the ground state. Such a process is sometimes called self-doping. [11, 10] On the language of energy levels a regime with the negative CT energy corresponds to the situation when the initial renormalized d−levels ǫd in (4) lie below the ligand p−levels. We start with an analysis of the covalent contribution to the CF splitting, Fig. 3(c). In contrast to the case of ∆CT > 0 it is now the lower bonding orbitals that have predominantly d−character with a small admixture of p−states ΨB = α|di + β|pi,

α2 + β 2 = 1,

α ≫ β.

(5)

For these bonding states the covalency leads to the splitting such that the t2g −levels lie above eg −ones, i.e. this covalency contribution leads to an “inverted” CF splitting for d−electrons.

Crystal field splitting in correlated systems with the negative charge transfer gap 10

∆(2+)

8 Energy (eV)

5

U ∆(3+)

6 4

∆(4+) 2 0 Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Figure 2. The charge-transfer (CT) gap ∆CT and the on-site Coulomb repulsion U values for different 3d-ions, after T. Mizokawa. [9]

Figure 3. (color online) Charge-transfer (CT) gap ∆CT < 0. The crystal-field splitting caused by the Coulomb interaction (b) and due to the covalency (c) are shown. Examples of the antibonding orbitals are sketched in the panel (a). Ligands forming octahedra are shown as open circles, d−orbitals are painted in blue and ligand p−orbitals in green. For the Coulomb contribution the charges (not the signs of the wave-functions) are indicated. The levels with predominantly d−character are marked by the dashed “box”.

These bonding states are actually completely filled. The electrons of partiallyfilled levels will actually be in the antibonding orbitals t∗2g , e∗g . Due to a stronger pdσ hybridization of orbitals having eg −symmetry with the ligand p−shell, the bondingantibonding splitting of eg −orbitals is larger than that of t2g −orbitals. As a result the order of antibonding levels is “normal”, the same as in Fig. 1 (t∗2g −levels are below ethe∗ −subshell). But, in contrast to the case of a positive CT gap, here the wave function g of the antibonding states will be ΨAB = α∗ |di + β ∗ |pi,

(α∗ )2 + (β ∗ )2 = 1,

α∗ ≪ β ∗ ,

(6)

i.e. these states have predominantly p−character. In effect the resulting state would be the state with the self-doping described above, i.e. with a large fraction of oxygen holes.

Crystal field splitting in correlated systems with the negative charge transfer gap

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The Coulomb contribution, however, would lead to the opposite sequence of d−levels, compared to the covalent term. For the d−levels themselves it would have the same sign as for the case of ∆CT > 0, and due to a stronger repulsion from ligands the eg −levels (bonding) would be pushed above t2g , opposite to the influence of the hybridization. One has to somewhat specify these arguments. In the case of ∆CT < 0 the ground state should be better taken not as dn p6 , but rather as dn+1 p5 , according to (2). This will change the magnitude of the Coulomb terms, the ting effect: still eg −electrons would repel stronger from the ligands (with smaller charge, e.g. O1.5− instead of O2− ). Thus in the case of the negative CT gap the contributions of the Coulomb interaction and of the p − d covalency to the CF splitting of d−states is opposite. The same can be said about the antibonding levels. As follows from Eq. (6), the antibonding levels have predominantly p−character, but the combinations of ligand p−electrons with eg -symmetry shown in Fig. 3(a), (e.g. the σ combination pσ = √14 [p1x − p2y − p3x + p4y ], which hybridizes with dx2 −y2 -orbital) would have stronger attraction to a positively-charged d−ion than the t∗2g −orbitals, made out of the π−combination of ligand orbitals. As a result the antibonding t∗2g −levels due to this factor would lie higher than e∗g (Fig. 3(b)). Thus we see that, in contrast to the “normal” case of positive CT gap, where both contributions to the CF splitting, the Coulomb contribution and the covalency, lead to qualitatively the same order of levels (antibonding eg -levels are above t2g , and vice versa for bonding states), for negative CT gaps the situation is different: these two contributions lead to the opposite level order. Therefore by studying the actual order of these levels, experimentally or theoretically, one can get some information about which of these contributions is in fact stronger. We undertook such a study on the example of a system with the negative CT gap - Cs2 Au2 Cl6 . 5. Crystal field in Cs2 Au2 Cl6 Cs2 Au2 Cl6 should formally contain the Au2+ (d9 ) ions. However, it is well-known in chemistry that this valence state of gold is unstable and is practically never observed experimentally: Au is known to exist in a valence Au1+ or Au3+ . This also happens in this material: there occurs a spontaneous charge disproportionation (CD) into Au1+ (Au2) and Au3+ (Au1), so that there appears in this compound two inequivalent Au positions, with checkerboard ordering in a cubic perovskite lattice. [12, 13] , see Fig. 4. In effect even the formula of this material is usually written not as CsAuCl3 , as for usual perovskites, but as Cs2 Au2 Cl6 = Cs2 Au1+ Au3+ Cl6 . We carried out the ab-initio band structure calculations of this material, have shown that it has negative CT gap, and analyzed the CF splitting. The ab-initio band-structure calculations of Cs2 Au2 Cl6 were performed within the frameworks of the Density Functional Theory (DFT) and the Generalized Gradient Approximation (GGA). We used the PW-SCF code with the ultrasoft version of

Crystal field splitting in correlated systems with the negative charge transfer gap

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Figure 4. The crystal structure of Cs2 Au2 Cl6 .

pseudopotentials [14, 15]. A maximum cut-off energy for the plane waves was chosen to be 200 eV. The Brillouin-zone (BZ) integration in the course of self-consistency interactions was performed over a mesh of 72 k-points. The structural data for ambient pressure were taken from Ref. [16]. The Wannier functions were constructed using the formalism described in Ref. [17]. -7

-6

-5

-4

-3

-2

-1

0

1

3

2

5

Au1

4

t2g

3

2

2

x -y

2 2

Partial DOS [states/(eV*atom)]

2

3z -r

1 0

Au2

4

t2g

3

2

2

x -y

2 2

2

3z -r

1

EF

0

Cl-p

4

Cl1-p DOS Cl2-p DOS

3 2 1 0

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Energy [eV]

Figure 5. Partial density of states for Cs2 Au2 Cl6 at ambient pressure. Au1 represent atom, which should have oxidation state 3+, while Au2 - 1+ (according to valence bond sum analysis of the crystal structure). The Cl−p partial density of states is the normalized per one Cl for two classes of “apical” Cl1 and “plane” Cl2 in the octahedra. The Fermi level is taken as zero.

The partial densities of states (PDOS) are presented in Fig. 5. One may clearly see

Crystal field splitting in correlated systems with the negative charge transfer gap

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Figure 6. The charge-density corresponding to the antibonding orbital of x2 − y 2 −symmetry centered on Au1 site, nominally Au3+ . The central part corresponds to the contribution of the real Au−5d x2 − y 2 orbital, while the rest is a σ combination of the Cl−p orbitals.

that indeed the lowest states in the valence band have largest contributions from the Au − d, not Cl − p states, so that the CT energy is indeed negative in this compound. The states closest to the top of the valence band, as well as the states in the conduction band, are mainly of p-character. The total occupation of Au2 5d−shell is 9.6, which is close to the d10 configuration, expected for Au1+ . However the number of d-electrons for Au1 is 9.3, much larger than 8 electrons expected from the naive ionic picture for the 3+ oxidation state of Au. Thus one should represent this states not as Au3+ (d8 ), but rather as d9 L or even d10 L2 , where L denotes ligand hole. This is due to the self-doping effect described above. Au3+ ions turned out to be Jahn-Teller active, but non-magnetic, with the configuration t62g e1g↑ e1g↓ (or rather d10 L ↑ L ↓). As it was shown in previous two sections, a larger hybridization leads to a larger bonding-antibonding splitting, independent on the sign of the ∆CT . The Cl octahedron surrounding Au1 is elongated along z−axis. Then the bonding-antibonding splitting for the orbital of x2 − y 2 -symmetry should be the largest among all of Au 5d−orbitals. This is exactly what we see in the upper panel of Fig. 5. The Coulomb contribution to the crystal-field splitting would stabilize just the opposite sequence of levels, see Fig. 3(b). This means that at least in this system the Coulomb contribution to the total value of the CF splitting is much less than the covalent term. It is important to mention that the conduction band is basically formed by the σ combination of Cl−p states with x2 − y 2 symmetry. These are actually the antibonding e∗g orbitals of the type (6), shown in Fig. 3(a). The charge density for this orbital, obtained using the Wannier function projection technique, is shown in Fig. 6. The detailed calculations show that about ∼20 % of the charge-density corresponding to this orbital has a 5d−character, while the main part ∼ 80 % comes from the Cl−p orbitals. A similar analysis can be performed for the compressed along the z-axis Au2O6

Crystal field splitting in correlated systems with the negative charge transfer gap

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octahedra. Here again the covalent contribution dominates and pushes down the bonding orbital of 3z 2 − r 2 -symmetry, which has the largest hybridization with the Cl−p states. The question which naturally arises is how general is the conclusion reached above, that the covalency contribution to the CF splitting is (much) larger than the Coulomb contribution. Generally speaking, one could think that the covalency contribution to the t2g − eg splitting, which is t2pdσ − t2pdπ , (7) ∼ ∆CT could become small for very large |∆CT |. It is not surprising that this contribution dominates for systems with small or slightly negative CT gap (apparently Cs2 Au2 Cl6 belongs to this class). Different approaches [7, 18], however, demonstrate that it is also the case for the values of parameters typical for “normal” TM compounds, such as KNiF3 [7] or NiO. [18] Nevertheless, one can not exclude that in some cases (e.g. for the early 3d metals) the situation could be different and the Coulomb contribution would become comparable or larger than the covalency one. ∆cov. CF

6. Summary In this paper we analyzed possible features of the crystal field splitting in different situations, and argued that for the negative charge-transfer gaps, in contrast to the “normal” situation with ∆CT > 0, different contributions to the crystal field splitting, the Coulomb contribution and that of the covalency with the ligands, act in opposite direction: the Coulomb contribution always pushes the t2g levels below the eg ones, whereas the p − d hybridization would lead to the opposite order of the bonding levels (which for negative charge transfer gap have predominantly d−character). The study of a concrete example of Cs2 Au2 Cl6 demonstrates that this system indeed has negative charge-transfer gap, so that the states in the conduction band are largely made of p−states of Cl. The bonding states of d−character lie ∼2-6 eV below the Fermi energy, and in this region the eg levels lie below t2g ones. Thus in this particular system the p − d hybridization dominates and determines the sequence of levels. We discuss in conclusion whether the opposite situation could be possible, in which case the sequence of d−levels would be determined by the Coulomb contribution and might be different from the usual one. 7. Acknowledgments We acknowledge fruitful communications with I. Mazin and M. Haverkort. We are very grateful to D. Korotin, for the code used for the projection on the Wannier functions, and especially to T. Mizokawa for providing us and allowing to use the Fig. 2. This work was supported by the German projects SFB 608, DFG GR 1484/2-1 and FOR 1346, by the European project SOPRANO, by Russian projects RFBR 10-02-00046

Crystal field splitting in correlated systems with the negative charge transfer gap

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and 10-02-96011, by the program of President of Russian Federation MK-309.2009.2, the Russian Federal Agency of Science and Innovation N 02.740.11.0217 and the scientific program “Development of scientific potential of universities” N 2.1.1/779. References [1] Zaanen J, Sawatzky G A and Allen J W 1985 Phys. Rev. Lett. 55 418 [2] Khomskii D I 1997 Lithuanian Journal of Physics 37, 2001 cond. mat. 0101164 [3] Often, e.g. in cuprates, one works in the hole representation, and then the energy of a transition (2) in this representation would be simply ∆CT = ǫ˜p−hole − ǫ˜d−hole ). [4] Harrison W A 1999 Elementary Electronic Structure (World Scientific, Singapore) [5] Kleiner W H 1952 J. Chem. Phys. 20 1784 [6] Tanabe Y and Sugano S 1956 J. Phys. Soc. Japan 11 864 [7] Sugano S and Shulman R G 1963 Phys. Rev. 130 517 [8] Bocquet A E, Mizokawa T, Saitoh T, Namatame H, and A. Fujimori A 1992 Phys. Rev. B 46 3771 [9] Mizokawa T Priv. Commun [10] Korotin M A, Anisimov V I, Khomskii D I and Sawatzky G A 1997 Phys. Rev. Lett. 80 4305 [11] Khomskii D I and Sawatzky G A 1997 Solid State Commun. 87 102 [12] Elliott N and Pauling S L 1938 J. Am. Chem. Soc. 60 1846 [13] Tindermanns J C M v. Eijndhoven and Verschoor S G C 1974 Mater. Res. Bull. 9 1667 [14] Vanderbilt D 1990 Phys. Rev. B. 41 7892 [15] Baroni S et al., PWscf (Plane-Wave Self-Consistent Field) codes, http://www.pwscf.org [16] Denner W, Schulz H, and D’Amour H 1979 Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr 35 360 [17] Korotin Dm, Kozhevnikov A V, Skornyakov S L, Leonov I, Bingelli N, Anisimov V I and Trimarchi G 2008 Eur. Phys. J. B 65 91 [18] Haverkort M et al. to be published